Can Momentum be Calculated Without Measuring Position in Quantum Mechanics?

[Curious6]

If we measure a particle's momentum don't we need to use some measure of the particle's position to find out the momentum ? How is momentum calculated if not through a measure somehow of the particle's position and how does this relate to the uncertainty principle?

 

[f95toli]

There are various ways to solve this experimentally.
However, the important point to realize here is that when we are talking about momentum in QM we are not always referring to particles (e.g. a particle with mass m travling at speed v); QM formalism such as HUP is VERY general and is valid for generalized position and momentum.
The word "generalized" here refers to the fact that we are talking about operators that have the same mathematical properties as position/momentum of a particles (the terms "generalized momentum/position" comes from Lagrangian mechanics) .
A specific example would be charge and phase in electrical systems (charge and phase do not commute so there is a HUP for them).

Hence, there isn't much point in trying to understand e.g. the HUP by referring to specific experiments in e.g. particle physics, optics and so on.

 

[ZapperZ]

If we measure a particle's momentum don't we need to use some measure of the particle's position to find out the momentum ? How is momentum calculated if not through a measure somehow of the particle's position and how does this relate to the uncertainty principle?

You have to detect the particle somehow and derive the necessary parameter. So yes, essentially, you are detecting where a particle hits a detector. But this "position" can corresponds to another quantity.

For example, after a particle passes through a single slit, it gains a transverse "momentum". That's why you see a large spread in the pattern on a screen. Where a particle hits the screen corresponds to the momentum it gained after it passes through the slit. So the particle position on the screen can be used to calculate the corresponding momentum.

Similar technique is used in http://tigger.uic.edu/~jcc/arpes.html" (ARPES) where the energy and in-plane momentum of the outgoing photoelectrons are measured simultaneously. The photoelectrons hit the detector at various locations depending on the polar angle that it was emitted from the photocathode material. So those locations give you the polar angle, which in turn, tells you the in-plane momentum. My avatar is one such example of an ARPES result.

Zz.

 

[Curious6]

You have to detect the particle somehow and derive the necessary parameter. So yes, essentially, you are detecting where a particle hits a detector. But this "position" can corresponds to another quantity.

For example, after a particle passes through a single slit, it gains a transverse "momentum". That's why you see a large spread in the pattern on a screen. Where a particle hits the screen corresponds to the momentum it gained after it passes through the slit. So the particle position on the screen can be used to calculate the corresponding momentum.

So, basically, if we can calculate the momentum by the particle position on the screen, we can calculate both position and momentum of a particle at any given time but in retrospect right?

Does this mean that a particle has both position and momentum at any given time but that observing position accurately reduces our ability to observe momentum accurately, and viceversa? Is this not an epistemological question in the sense that this position and momentum cannot be known at same time, but ontologically, particles do have position and momentum at same time.

 

[ZapperZ]

So, basically, if we can calculate the momentum by the particle position on the screen, we can calculate both position and momentum of a particle at any given time but in retrospect right?

No, you have to be VERY careful of what "position" and what "momentum" you are calculating. This is because it depends on the setup and the experiment.

The position that is on the detector, and the momentum that one measures based on THAT position is NOT the "x" and "p" that are in the HUP. Here, the position corresponds to the momentum. If there's any position operator being measured, it must have been measured before this (such as a single slit that defines the position of the particle that passed through it.

Zz.